In a famous paper published in 1904, Helge von Koch introduced the curve that still serves nowadays as an iconic representation of fractal shapes. In fact, von Koch's main goal was the construction of a continuous but nowhere differentiable function, very similar to the snowflake, using elementary geometric procedures, and not analytical formulae. We prove that a parametrized family of functions (including and) generalizing von Koch's example enjoys a rich multifractal behavior, thus enriching the class of historical mathematical objects having surprising regularity properties. The analysis relies on the study of the orbits of an underlying dynamical system and on the introduction of self-similar measures and non-trivial iterated functions systems adapted to the problem.

In this article, we investigate the bivariate multifractal analysis of pairs of Borel probability measures. We prove that, contrarily to what happens in the univariate case, the natural extension of the Legendre spectrum does not yield an upper bound for the bivariate multifractal spectrum. For this we build a pair of measures for which the two spectra have disjoint supports. Then we study the bivariate multifractal behavior of an archetypical pair of randomly correlated measures, which give new, surprising, behaviors, enriching the narrow class of measures for which such an analysis is achieved.

Given a Radon probability measure μ supported in ℝ^d , we are interested in those points x around which the measure is concentrated infinitely many times on thin annuli centered at x . Depending on the lower and upper dimension of μ , the metric used in the space and the thinness of the annuli, we obtain results and examples when such points are of μ -measure 0 or of μ -measure 1. The measure concentration we study is related to “bad points” for the Poincaré recurrence theorem and to the first return times to shrinking balls under iteration generated by a weakly Markov dynamical system. The study of thin annuli and spherical averages is also important in many dimension-related problems, including Kakeya-type problems and Falconer’s distance set conjecture.

In this work and its companion [1], we construct Baire function spaces in which typical elements share the same prescribed multifractal behavior and obey a multifractal formalism, providing a solution to the so-called Frisch-Parisi conjecture for functions, an inverse problem raised by S. Jaffard. In this first part, a family Ed of almost-doubling fully supported capacities on Rd with prescribed singularity spectra is constructed. With each & mu; & ISIN; Ed we associate a Baire function space B & mu;(Rd) (a generalisation of Holder-Zygmund spaces) in which typical functions share the same singularity spectrum as & mu;. This yields a partial solution to the conjecture. In [1], we introduce and study a family B = {B & mu;,p q (Rd)}& mu;EEd,(p,q)E[1,+oo]2 of heterogeneous Besov spaces that contains {B & mu;(Rd)}& mu;EEd and generalises in a natural direction the family of standard Besov spaces, and we solve the inverse problem exhaustively inside B. & COPY; 2023 Elsevier Masson SAS. All rights reserved.

In this paper, we determine the almost sure multifractal spectrum of a class of random functions constructed as sums of pulses with random dilations and translations. In addition, the continuity modulii of these functions is investigated.

In this article, starting from a Gibbs capacity, we build a new random capacity by applying two simple operators, the first one introducing some redundancy and the second one performing a random sampling. Depending on the values of the two parameters ruling the redundancy and the sampling, the new capacity has very different multifractal behaviors. In particular, the multifractal spectrum of the capacity may contain two to four phase transitions, and the multifractal formalism may hold only on a strict subset (sometimes, reduced to a single point) of the spectrum's domain.

The macroscopic Hausdorff dimension DimH(E) of a set E⊂Rd was introduced by Barlow and Taylor to quantify a “fractal at large scales” behavior of unbounded, possibly discrete, sets E. We develop a method based on potential theory in order to estimate this dimension in Rd. Then, we apply this method to obtain Marstrand-like projection theorems: given a set E⊂R2, for almost every θ∈[0,2π], the projection of E on the straight line passing through 0 with angle θ has dimension equal to min⁡(DimH(E),1).

Frisch-Parisi conjecture claims the existence of Baire function spaces in which Baire typical functions share the same multifractal behavior, prescribed in advance, and obey a multifractal formalism. In this paper, we introduce a family B of heterogeneous Besov spaces, which generalize the standard Besov spaces -they are obtained by replacing the Lebesgue measure (which plays a key role in the definition of the standard Besov spaces) by multifractal Radon measures belonging to some class constructed in the companion paper [1]. We find a characterization of the elements of B in terms of wavelet coefficients, and then describe the multifractal properties (singularity spectrum, validity of the multifractal formalism) of their Baire typical functions. This allows us to fully solve the Frisch-Parisi conjecture inside B.& COPY; 2023 Elsevier Masson SAS. All rights reserved.

Multifractal behavior has been identified and mathematically established for large classes of functions, stochastic processes and measures. Multifractality has also been observed on many data coming from Geophysics, turbulence, Physics, Biology, to name a few. Developing mathematical models whose scaling and multifractal properties fit those measured on data is thus an important issue. This raises several still unsolved theoretical questions about the prescription of multifractality (i.e. how to build mathematical models with a singularity spectrum known in advance), typical behavior in function spaces, and existence of solutions to PDEs or SPDEs with possible multifractal behavior. In this survey, we gather some of the latest results in this area. Dedicated to Alain Arn{\'e}odo, pioneer in the development of wavelet tools for data analysis.

For any s ∈ (1/2, 1], the series Fs(x) = ∑∞ n=1 e iπn2x/ns converges almost everywhere on [−1, 1] by a result of Hardy-Littlewood concerning the growth of the sums ∑N n=1 e iπn2x, but not everywhere. However, there does not yet exist an intrinsic description of the set of convergence for Fs. In this paper, we define in terms of even continued fractions a subset of points of [−1, 1] of full measure where the series converges. As an intermediate step, we prove that, for s > 0, the sequence of functions N ∑ n=1 e x ns − eπ4 |x| 1 2 ⌊N |x|⌋ ∑ n=1 e /x ns converges when N → ∞ to a function Ωs continuous on [−1, 1] \ {0} with (at most) a singularity at x = 0 of type x s−1 2 (s 6= 1) or a logarithmic singularity (s = 1). We provide an explicit expression for Ωs and the error term. Finally, we study thoroughly the convergence properties of certain series defined in term of the convergents of the even continued fraction of an irrational number.

Let $\unicode[STIX]{x1D707}$ be the projection on $[0,1]$ of a Gibbs measure on $\unicode[STIX]{x1D6F4}=\{0,1\}^{\mathbb{N}}$ (or more generally a Gibbs capacity) associated with a Hölder potential. The thermodynamic and multifractal properties of $\unicode[STIX]{x1D707}$ are well known to be linked via the multifractal formalism. We study the impact of a random sampling procedure on this structure. More precisely, let $\{{I_{w}\}}_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ stand for the collection of dyadic subintervals of $[0,1]$ naturally indexed by the finite dyadic words. Fix $\unicode[STIX]{x1D702}\in (0,1)$, and a sequence $(p_{w})_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ of independent Bernoulli variables of parameters $2^{-|w|(1-\unicode[STIX]{x1D702})}$. We consider the (very sparse) remaining values $\widetilde{\unicode[STIX]{x1D707}}=\{\unicode[STIX]{x1D707}(I_{w}):w\in \unicode[STIX]{x1D6F4}^{\ast },p_{w}=1\}$. We study the geometric and statistical information associated with $\widetilde{\unicode[STIX]{x1D707}}$, and the relation between $\widetilde{\unicode[STIX]{x1D707}}$ and $\unicode[STIX]{x1D707}$. To do so, we construct a random capacity $\mathsf{M}_{\unicode[STIX]{x1D707}}$ from $\widetilde{\unicode[STIX]{x1D707}}$. This new object fulfills the multifractal formalism, and its free energy is closely related to that of $\unicode[STIX]{x1D707}$. Moreover, the free energy of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ generically exhibits one first order and one second order phase transition, while that of $\unicode[STIX]{x1D707}$ is analytic. The geometry of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ is deeply related to the combination of approximation by dyadic numbers with geometric properties of Gibbs measures. The possibility to reconstruct $\unicode[STIX]{x1D707}$ from $\widetilde{\unicode[STIX]{x1D707}}$ by using the almost multiplicativity of $\unicode[STIX]{x1D707}$ and concatenation of words is discussed as well.

In this article, a solution to the so-called Frisch-Parisi conjecture is brought. This achievement is based on three ingredients developed in this paper. First almost-doubling fully supported Radon measures on ^d with a prescribed singularity spectrum are constructed. Second we define new heterogeneous Besov spaces B^μ,p_q and find a characterization using wavelet coefficients. Finally, we fully describe the multifractal nature of typical functions in the function spaces B^μ,p_q. Combining these three results, we find Baire function spaces in which typical functions have a prescribed singularity spectrum and satisfy a multifractal formalism. This yields an answer to the Frisch-Parisi conjecture.

We describe the size of the sets of sojourn times $E_{\gamma }=\{t\geq 0:|B_{t}|\leq t^{\gamma }\}$ associated with a fractional Brownian motion $B$ in terms of various large scale dimensions.

We investigate the large scale structure of certain sojourn sets of one dimensional Brownian motion within two-sided moving boundaries. The macroscopic Hausdorff dimension, upper mass dimension and logarithmic density of these sets are computed. We also give a uniform macroscopic dimension result for the Brownian level sets.

Multifractal analysis, that quantifies the fluctuations of regularities in time series or textures, has become a standard signal/image processing tool. It has been successfully used in a large variety of applicative contexts. Yet, successes are confined to the analysis of one signal or image at a time (univariate analysis). This is because multivariate (or joint) multifractal analysis remains so far rarely used in practice and has barely been studied theoretically. In view of the myriad of modern real-world applications that rely on the joint (multivariate) analysis of collections of signals or images, univariate analysis constitutes a major limitation. The goal of the present work is to theoretically ground multivariate multifractal analysis by studying the properties and limitations of the most natural extension of the univariate formalism to a multivariate formulation. It is notably shown that while performing well for a class of model processes, this natural extension is not valid in general. Based on the theoretical study of the mechanisms leading to failure, we propose alternative formulations and examine their mathematical properties.

Multivariate multifractal analysis proposes to estimate the multivariate multifractal spectrum of several signals as a way to reveal the correlations between their singularity sets. Recently discovered failures of the "natural" multifractal formalism challenge its foundations, and raise questions on the information it provides. We investigate these questions by supplying some general results on the multivariate multifractal analysis of processes satisfying the large intersection property, and illustrate it on random lacunary wavelet series.

We show how a joint multifractal analysis of a collection of signals unravels correlations between the locations of their pointwise singularities. The multivariate multifractal formalism, reformulated in the general setting supplied by multiresolution quantities, provides a framework which allows to estimate joint multifractal spectra. General results on joint multifractal spectra are derived, and illustrated by the theoretical derivation and practical estimation of the joint multifractal spectra of simple mathematical models, including correlated binomial cascades.

With the aim of surgical success, the evaluation of dental implant long-term stability is an important task for dentists. About that, the complexity of the newly formed bone and the complex boundary conditions at the bone-implant interface induce the main difficulties. In this context, for the quantitative evaluation of primary and secondary stabilities of dental implants, ultrasound based techniques have already been proven to be effective. The microstructure, the mechanical properties and the geometry of the bone-implant system affect the ultrasonic response. The aim of this work is to extract relevant information about primary stability from the complex ultrasonic signal obtained from a probe screwed to the implant. To do this, signal processing based on multiscale analysis has been used. The comparison between experimental and numerical results has been carried out, and a correlation has been observed between the multifractal signature and the stability. Furthermore, a sensitivity study has shown that the variation of certain parameters (i.e. central frequency and trabecular bone density) does not lead to a change in the response.

Let y be an irreducible topological Markov shift, and let mu be a shift-invariant Gibbs measure on y. Let (X-n)(n >= 1) be a sequence of i.i.d. random variables with common law mu. In this paper, we focus on the size of the covering of y by the balls B(X-n,n(-s)). This generalises the original Dvoretzky problem by considering random coverings of fractal sets by non-homogeneously distributed balls. We compute the almost sure dimension of lim sup(n ->+infinity) B(X-n, n(-s)) for every s >= 0, which depends on s and the multifractal features of mu. Our results include the inhomogeneous covering of T-d and Sierpinski carpets.